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Distributed and Real-time Predictive Control
Melanie Zeilinger
Christian Conte (ETH) Alexander Domahidi (ETH)
Ye Pu (EPFL) Colin Jones (EPFL)
• Large-scale, complex system • Constraints • Uncertainties • High performance and safety
• Composed of coupled subsystems • Often high-speed dynamics • Computation and communication
constraints
Challenges in modern control systems
Courtesy of
Customer: - Control of building
networks - Control of flexible loads
and storage capacities
Power system: - Frequency control - Voltage control
Electric vehicles
i4Energy seminar: 12pm, 310 SDH "The Role of Supply-Following Loads in Highly Renewable Electricity Grids”, Jay Taneja
2
• Large-scale, complex system • Constraints • Uncertainties • High performance and safety
• Composed of coupled subsystems • Often high-speed dynamics • Computation and communication
constraints
Challenges in modern control systems
Electric vehicles
Courtesy of
Power network Traffic network
Courtesy of Dr. Pu Wang
Robotics
Courtesy of IDSC, ETH
3
Model Predictive Control (MPC) "– A High Performance Method for Constrained Control
state x
output y system
Each sample time: 1. Measure / estimate state 2. Solve optimization problem for entire planning window 3. Implement only the first control action
u�(x) := argmin Vf (xN) +N�1�
k=0
l(xk , uk)
Z�[� x0 = x TLHZ\YLTLU[xk+1 = f (xk , uk) Z`Z[LT TVKLS(xk , uk) � X � U JVUZ[YHPU[ZxN � Xf PU]HYPHUJL
x!1
x!4
x!0 = x
x!5
Xfx!2
x!3
4
Model Predictive Control (MPC) "– A High Performance Method for Constrained Control
state x
output y system
Established approach: • Optimality • Terminal cost and constraint
Classical MPC theory: J High performance J Recursive constraint satisfaction J Stability by design
u�(x) := argmin Vf (xN) +N�1�
k=0
l(xk , uk)
Z�[� x0 = x TLHZ\YLTLU[xk+1 = f (xk , uk) Z`Z[LT TVKLS(xk , uk) � X � U JVUZ[YHPU[ZxN � Xf PU]HYPHUJL
5
Real-time Model Predictive Control
state x
output y system
u�(x) := argmin Vf (xN) +N�1�
k=0
l(xk , uk)
Z�[� x0 = x TLHZ\YLTLU[xk+1 = f (xk , uk) Z`Z[LT TVKLS(xk , uk) � X � U JVUZ[YHPU[ZxN � Xf PU]HYPHUJL
Embedded processor
Bounded computation time à Early termination à Invalidates MPC theory based on optimality
Classical MPC theory: J High performance J Recursive constraint satisfaction J Stability by design
6
Distributed Model Predictive Control
Local computation and information: à Restrictive local terminal conditions à Stability in exchange for significant
conservatism
… contr1
sys1
contr3
sys3 …
…
sys2
contr2
contrM
sysM
Classical MPC theory: J High performance J Recursive constraint satisfaction J Stability by design
7
Outline: Distributed and Real-time MPC Established approach: • Optimality • Terminal cost and constraint
Centralized MPC theory: J Recursive constraint satisfaction J Stability by design
Outline (Part II): Stability with larger region of attraction based on local information à Plug and Play MPC
Distributed MPC: • Reduced conservatism through
distributed optimization • BUT: Global terminal conditions
Outline (Part I): Stability and constraint satisfaction for any real-time constraint à MPC for fast, safety-critical systems
Real-time MPC: • Flexibility and fast convergence
through interior-point methods • BUT: Variable solve-times
8
Outline: Distributed and Real-time MPC Established approach: • Optimality • Terminal cost and constraint
Centralized MPC theory: J Recursive constraint satisfaction J Stability by design
Outline (Part II): Stability with larger region of attraction based on local information à Plug and Play MPC
Distributed MPC: • Reduced conservatism through
distributed optimization • BUT: Global terminal conditions
Outline (Part I): Stability and constraint satisfaction for any real-time constraint à MPC for fast, safety-critical systems
Real-time MPC: • Flexibility and fast convergence
through interior-point methods • BUT: Variable solve-times
9
Stability and Invariance of Optimal MPC
Assumptions: 1. Xf ⇢ X is invariant
x 2 Xf ) Ax + Buf (x) 2 Xf
2. Vf (x) is a Lyapunov function in XfVf (Ax + Buf (x))� Vf (x) �l(x, uf (x))
V �N (x) = min VN(x,u) := Vf (xN) +N�1�
i=0
xTi Qxi + uTi Rui
Z�[� x0 = x TLHZ\YLTLU[xi+1 = Axi + Bui Z`Z[LT�TVKLSCxi +Dui � b JVUZ[YHPU[ZxN � Xf [LYTPUHS�JVUZ[YHPU[
x!0 = x
x!4
x!5
Xf
x!1 = x+
x!2
x!3
Ax�5 + Buf (x�5 )
Theorem: V �N (x)• is a convex Lyapunov function
• The feasible set is invariant under the optimal MPC controller
10
Stability and Invariance of Optimal MPC
Assumptions: 1. Xf ⇢ X is invariant
x 2 Xf ) Ax + Buf (x) 2 Xf
2. Vf (x) is a Lyapunov function in XfVf (Ax + Buf (x))� Vf (x) �l(x, uf (x))
x!0 = x
x!4
x!5
Xf
x!1 = x+
x!2
x!3
Ax�5 + Buf (x�5 )
Proof: Shifted sequence • is feasible à Recursive feasibility and invariance • decreases the cost
à is a Lyapunov function V �N (x
+)� V �N (x) � V ZOPM[N (x+)� V �N (x) � �l(x, u�0) < 0
V �N (x)
uZOPM[ = [u!1, . . . , u!N"1, Kx
!N ]
V �N (x) = min VN(x,u) := Vf (xN) +N�1�
i=0
xTi Qxi + uTi Rui
Z�[� x0 = x TLHZ\YLTLU[xi+1 = Axi + Bui Z`Z[LT�TVKLSCxi +Dui � b JVUZ[YHPU[ZxN � Xf [LYTPUHS�JVUZ[YHPU[
11
Real-Time MPC Controller Synthesis
Ideal approach is problem specific
Interior point methods • Modify controller to
be robust to time constraints
Large-scale, ms Gradient approaches • Bound computation
time a priori
Medium-scale, us Pre-compute controller • Fixed time online
Small-scale, ns
Generic optimization
code Deterministic
optimizer Analytic
expression for control law
x x x u u
Flexibility
Speed
u
12
Real-time MPC using interior-point methods Real-time online MPC: Guarantee that • within the real-time constraint • a feasible solution • satisfying stability criteria • for any admissible initial state is found.
13
Real-time MPC using interior-point methods Real-time online MPC: Guarantee that • within the real-time constraint ⇐ Early termination • a feasible solution ⇐ Warm-start • satisfying stability criteria • for any admissible initial state is found.
Suboptimal solution
Warm-start Online optimization
x+
x
14
Real-time MPC using interior-point methods Real-time online MPC: Guarantee that • within the real-time constraint ⇐ Early termination • a feasible solution ⇐ Warm-start • satisfying stability criteria • for any admissible initial state is found.
Many recent codes have demonstrated that extreme speeds are possible…
qpOases Online Active Set Strategy
CVXGEN Code Generation for Convex Optimization
QPSchur A dual, active-set, Schur-complement method for quadratic programming
OOQP Object-oriented software for quadratic programming
… but cannot guarantee stability in a real-time setting!
15
-4 -3 -2 -1 0 -0.5
0
0.5
1
1.5
2
x 2
x 1
Example: Effect of limited computation time
Closed loop trajectory: Optimal control law
0 5 10 15 20 25 1
2
3 x 10 -4
Time Step
Com
puta
tion
time
[s]
Closed loop trajectory: Optimization stopped after 4 iterations
= max computation time of 21ms
Limited computation time => No stability properties
Unstable example
x+ =
�1.2 10 1
�x +
�10.5
�u
|x1| � 5,�5 � x2 � 1
|u| � 1, N = 5, Q = I, R = 1
16
-4 -3 -2 -1 0 -0.5
0
0.5
1
1.5
2
x 2
Real-time robust MPC : Nearly optimal and satisfies time constraints
Example: Stability under proposed real-time method
Proposed real-time MPC method"
stopped after 4 online iterations
Closed loop trajectory: Optimal control law
0 5 10 15 20 25 1
2
3 x 10 -4
Time Step
Com
puta
tion
time
[s]
x 1 Closed loop trajectory:
Optimization stopped after 4 iterations = max computation time of 21ms
Unstable example
x+ =
�1.2 10 1
�x +
�10.5
�u
|x1| � 5,�5 � x2 � 1
|u| � 1, N = 5, Q = I, R = 1
17
Loss of stability guarantee in real-time Requirement for stability: Lyapunov function
à Use of MPC cost as Lyapunov function à Key condition: Decrease of MPC cost at every time step
Using interior-point methods this condition can be violated even when initializing with a stabilizing sequence, e.g. the shifted sequence
Example: Barrier interior-point method Minimize augmented cost
z* z*(10) minz
f (z)
Z�[� Fz = Ex
Gz � d
minz
f (z)� µm�
i=1
log(�Giz + di)
Z�[� Fz = Ex
à Decrease in augmented cost does not enforce a decrease in MPC cost à Steady-state offset for μ≠0
VN(xt ,ut) < VN(xt!1,ut!1)
18
Suboptimal cost for any feasible solution to real-time problem provides Lyapunov function
Real-time stability guarantees Goal: Ensure that suboptimal cost is Lyapunov function
Introduce ‘Lyapunov constraint’: Enforces decrease in suboptimal MPC cost at each iteration
If … • We can provide (strictly) feasible solution for Lyapunov constraint in real-time Key: Ensure that epsilon progress is always possible without optimization à Technique based on warm-starting from previous sampling time!• We can solve quadratically constrained QPs with modified structure
(Quadratic constraint)
Theorem:
VN(xUVTt ,ut) ! VN(xt!1,ut!1)" !#xt!1#2Q
à Stability for any real-time constraint
19
Computation times on Intel Atom for QP
0 5 10 15 20 25 3010−5
10−4
10−3
10−2
10−1
77 μs
Time per"iteration (s)
Number of masses (Number of states/2)
FORCES CVXGEN
Oscillating masses:
QP with - box constraints - diagonal cost
More details in [Domahidi, et al., ACC 2012]. forces.ethz.ch 20
Computation times on Intel Atom for QP
0 5 10 15 20 25 3010−5
10−4
10−3
10−2
10−1
Time per"iteration (s)
Number of masses (Number of states/2)
FORCES Oscillating masses:
QP with - box constraints - diagonal cost
QCQP with - quadr. terminal set - real-time constr.
FORCES RT
207 μs
0 5 10 15 20 25 30
10−4
10−3
10−2
10−1
FORCES RT is stabilizing for "all numbers of iterations! ⇒ 207 μs is obtainable ⇒ Other methods ~10 iterations
CVXGEN
More details in [Domahidi, et al., ACC 2012]. forces.ethz.ch 21
Summary: Real-Time MPC
• Optimal MPC requires unknown computation time à Fast systems require theory of real-time MPC • Real-time method provides stability guarantees for arbitrary time constraints • Extension to robust tube-based MPC • Extension to tracking (more involved) • Possible to achieve millisecond solve-times on inexpensive hardware • Real-time MPC still faster than solvers without guarantees
[Zeilinger, et al., Automatica 2013, accepted], [Domahidi, et al., CDC 2012]
Real-time online MPC: Guarantee that • within the real-time constraint ⇐ Early termination • a feasible solution ⇐ Warm-start • satisfying stability criteria ⇐ Lyapunov constraint • for any admissible initial state is found.
22
Outline: Distributed and Real-time MPC Established approach: • Optimality • Terminal cost and constraint
Centralized MPC theory: J Recursive constraint satisfaction J Stability by design
Outline (Part II): Stability with larger region of attraction based on local information
Distributed MPC: • Reduced conservatism through
distributed optimization • BUT: Global terminal conditions
Outline (Part I): Stability and constraint satisfaction for any real-time constraint
Real-time MPC: • Flexibility and fast convergence
through interior-point methods • BUT: Variable solve-times
23
Distributed Model Predictive Control (MPC)
… contr1
sys1
contr3
sys3 …
…
sys2
contr2
contrM
sysM
Independent constraints xi � Xi ui � Ui
Coupled linear dynamics x+i =
!Mj=1 Ai jxj + Biui = ANi xNi + Biui
Communication with neighbours Ni
How to ensure stability and constraint satisfaction without central coordination?
Cooperative objective l(x, u) =
!Mi=1 li(xi , ui)
24
Modified dynamics
Distributed Model Predictive Control (MPC)
… contr1
sys1
contr3
sys3 …
…
sys2
contr2
contrM
sysM
sys4
contr4
Plug and Play MPC: Allow subsystems to join or leave the network
How to maintain stability and constraint satisfaction during network changes?
x+i =!j!NTVK
iAi jxj + Biui
25
Example: Dual Decomposition Gradient of the dual function:
Distributed Optimization Requires Structure
min�
fi(yi)
Z�[� yi � Yi�
Aiyi = c
Distributed optimization requires that the problem is
structured
Gradient-based approach Optimal values yi* ➙ Local optimization"
Dual update ➙ Consensus
Many variants on this theme (ADMM, AMA,...)
g(�) = minyi�Yi
�fi(yi) + �T
��Aiyi � c
�=
�minyi�Yi
fi(yi) + �TAiyi
�+ = �+ ��g(�)
�g(�) =�
Aiy�i (�)� c
26
Two Conflicting Requirements
Structured optimization
Stability and invariance if:
min Vf (xN) +N�1X
i=0
l(xi , ui)
s.t. x0 = x
xi+1 = Axi + Bui
(xi , ui) 2 X ⇥ UxN 2 Xf
Plant
A, B structured
X , U distributed
l(x, u) distributed
1 2 Terminal cost & constraints:
Xf = X 1f � · · ·� Xf
Vf (x) =M�
k=1
V kf (xNk )
uf (x) = [u1f (xN1), . . . , uMf (xNM )]
T
1. Xf � X is invariantx � Xf � Ax + Buf (x) � Xf
2. Vf (x) is a Lyapunov function in Xf
Vf (Ax + Buf (x))� Vf (x) � �l(x, uf (x))
Goal: Satisfy both requirements without central coordination à Online & offline optimization structured according to system coupling
Dense
Dense
27
Structured Lyapunov Function Lyapunov requirement:
Structure requirement: Vf (x) = V 1f (x1) + · · ·+ V Mf (xM)Vf (x
+)� Vf (x) � �l(x, uf (x))
Theorem: [Jokic, Lazar, 2009]
Vf (x) :=M�
i=1
V if (xNi ) is a Lyapunov function if
V if (x
+i )� V i
f (xi) � �li(xNi ) + �i(xNi )
M�
i=1
�i(xNi ) � 0
Possible local increase
Global decrease
J Global Lyapunov function à Stability
Idea: Allow local increase while requiring a global decrease
28
Structured Invariant Set
Idea: Level sets of a Lyapunov function are invariant
Want a condition that can be tested in a distributed fashion
V if (x
+i )� V i
f (xi) � �li(xNi , uif (xNi )) + �i(xNi ) �� 0
Xf =
�
x
����� Vf (x) =M�
i=0
V if (xNi ) � �
�
Problem: This terminal constraint couples all sub-systems
7YVISLT! :[H[PJ ZL[Z X if (�i) HYL UV[ PU]HYPHU[���
Invariance requirement:
Structure requirement: Xf (�) = X 1f (�1)� · · ·� XM
f (�M)
x � Xf � x+ � Xf
Vf (xi) � �i �� Vf (x+i ) � �i � ZPUJL
X if (�i) = {x | V i
f (xNi ) � �i} whereM�
i=0
�i = � � �
29
Theorem:
Proof: From
1.
2.
Structured Dynamic Invariant Set Invariance requirement:
Structure requirement: Xf (�) = X 1f (�1)� · · ·� XM
f (�M)
x � Xf � x+ � Xf
• Define auxiliary dynamics, with the same structure as the system dynamics:
• Choose initial
1. Time-varying terminal set is invariant
2. All state and input constraints are satisfied in
X if (�i) = {x | V i
f (xi) � �i}
�+i = �i + �i(xNi )
Xf (�)
xi � X if (�i)� x+i � X
if (�
+i )
V if (x
+i ) � V i
f (xi)� li(xNi , uif (xNi )) + �i(xNi ) � �i + �i(xNi ) = �+i
Xf (�) � X � Xf (�+) � X
�i Z\JO [OH[��i � ��
�x
�� �V i
f (xNi ) � ��� X
��+i =
��i +
��i(xNi ) �
��i
30
Theorem:
Structured Dynamic Invariant Set Invariance requirement:
Structure requirement: Xf (�) = X 1f (�1)� · · ·� XM
f (�M)
x � Xf � x+ � Xf
• Define auxiliary dynamics, with the same structure as the system dynamics:
• Choose initial
1. Time-varying terminal set is invariant
2. All state and input constraints are satisfied in
X if (�i) = {x | V i
f (xi) � �i}
�+i = �i + �i(xNi )
Xf (�)
xi � X if (�i)� x+i � X
if (�
+i )
�i Z\JO [OH[��i � ��
�x
�� �V i
f (xNi ) � ��� X
J Recursive feasibility
31
Distributed MPC – Online Control
+PZ[YPI\[LK JVU[YVS �VUSPUL MVY L]LY` Z\IZ`Z[LT�!
�� 4LHZ\YL Z[H[L
�� :VS]L NSVIHS 47* WYVISLT I` KPZ[YPI\[LK VW[PTPaH[PVU� HWWS` PUW\[ ui
�� <WKH[L �+i = �i + �(xNi )
minM�
i=1
V if (xi(N)) +
N�1�
k=0
l(xi(k), ui(k))
s.t. xi(0) = xi
xi(k + 1) = Ai ixi(k) + Biui(k) +�
j�Ni
Ai jxj(k)
(xi(k), ui(k)) � X i � U i
xi(N) � X if (�i)
Structured MPC problem
32
Distributed MPC - Synthesis and Online Control
No central coordination required!
+PZ[YPI\[LK JVU[YVS �VUSPUL MVY L]LY` Z\IZ`Z[LT�!
�� 4LHZ\YL Z[H[L
�� :VS]L NSVIHS 47* WYVISLT I` KPZ[YPI\[LK VW[PTPaH[PVU� HWWS` PUW\[ ui
�� <WKH[L �+i = �i + xTNi(N)�Ni xNi (N)
33
+PZ[YPI\[LK Z`U[OLZPZ PU [OL SPULHY X\HKYH[PJ JHZL �VMÅPUL�!
�� :VS]L KPZ[YPI\[LK 340 [V JVTW\[L!� 3VJHS YLSH_LK 3`HW\UV] M\UJ[PVUZ V f
i (xi) = xTi Pixi
� 0UKLÄUP[L JV\WSPUN �i(xNi ) = xTNi�ixNi
� 3VJHS SPULHY JVU[YVS SH^Z ufi (xNi ) = KNi xNi
�� :VS]L KPZ[YPI\[LK 37 [V JVTW\[L PUP[PHS MLHZPISL [LYTPUHS ZPaL �
Computational example • Chain of inverted pendulums (unstable) • Linearized around the origin • States: Angle and angular velocity of each pendulum • Inputs: Torque at each pivot
34
Computational example – Closed-Loop Simulation
• 5 Pendulums, alternating direction method of multipliers, 100 iterations. • Initially all pendulums in origin, only pendulum 1 is deflected. • Cost of proposed method only 4% higher than centralized MPC and 21%
lower than for a trivial terminal set.
5 10 15 20 25 30 35 40 45 50−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Simulation step
Angl
e � i
5 10 15 20 25 30 35 40 45 50−10
−8
−6
−4
−2
0
2
4
Simulation step
Inpu
t Ti
Pend.1, centr. MPC (cost 9.021)Pend.2 centr. MPC (cost 9.021)Pend.1, distr. MPC (cost 9.360)Pend.2, distr. MPC (cost 9.360)Pend. 1, distr. MPC Xf=0 (cost 11.891)Pend. 2, distr. MPC Xf=0 (cost 11.891)
35
Computational example – Local Terminal Sets
Sizes of local terminal sets change dynamically
36
Computational example – Region of Attraction
• Maximum feasible deflection of the first pendulum vs. prediction horizons
• Short prediction horizons: Region of attraction for proposed method significantly larger than for trivial terminal set
• Long prediction horizons: All methods converge to the same maximum control invariant set
5 10 15 20 25 300
0.5
1
1.5
2
2.5
MPC prediction horizon
φ 1,m
ax
Centr. MPCDistr. MPCDistr. MPC Xf = 0
37
Distributed Model Predictive Control (MPC)
… contr1
sys1
contr3
sys3 …
…
sys2
contr2
contrM
sysM
sys4
contr4
Plug and Play MPC: Allow subsystems to join or leave the network
Maintain stability and recursive feasibility during network changes: • Adapt local control laws of subsystems and neighbours • Ensure feasibility of the modified control laws
C! Z`Z[LTZ [V IL YLKLZPNULKR! Z`Z[LTZ [OH[ YLTHPU \UJOHUNLK
Modified dynamics x+i =
!j!NTVK
iAi jxj + Biui
38
Preparation for Plug and Play operation:
Redesign Phase: Adapt local control laws of subsystems and neighbours
• Compute new local terminal costs and constraint sets for (virtually) modified network
V if (xi) X i
f (�i)
Transition Phase: Ensure feasibility of the modified MPC problem
• Compute a steady-state for Plug and Play operation such that: - Steady-state is a feasible initial state for the modified MPC problem - System can be controlled to the steady-state from the current state
• If steady-state can be found - Control system to steady-state - Permit plug and play operation
Else - Reject plug and play operation
Plug and Play operation = subsystems join/leave network and modified local control law is applied
… contr1
sys1
contr3
sys3 …
…
sys2
contr2
contrM
sysM
sys4
contr4
39
Preparation for Plug and Play operation:
Redesign Phase: Adapt local control laws of subsystems and neighbours
• Compute new local terminal costs and constraint sets for (virtually) modified network
V if (xi) X i
f (�i)
Transition Phase: Ensure feasibility of the modified MPC problem
• Compute a steady-state for Plug and Play operation such that: - Steady-state is a feasible initial state for the modified MPC problem - System can be controlled to the steady-state from the current state
• If steady-state can be found - Control system to steady-state - Permit plug and play operation
Else - Reject plug and play operation
Plug and Play operation = subsystems join/leave network and modified local control law is applied
… contr1
sys1
contr3
sys3 …
…
sys2
contr2
contrM
sysM
sys4
contr4
Plug and and play synthesis and control
via distributed optimization 40
Computational example – Area Generation Control
1"
2" 3"
4"
5"
• Four power generation areas with load frequency control • Model linearized around equilibrium (Saadat, 2002; Riverso, et al. 2012) Goals: - Restore frequency, follow load change - Allow fifth area to join the network
0 20 40 60 80!0.01
0
0.01
x[1]
2
t [s]
Area 1
0 20 40 60 80
!5
0
5
x 10!3
x[2]
2
t [s]
Area 2
0 20 40 60 80
!5
0
5
x 10!3
x[3]
2
t [s]
Area 3
0 20 40 60 80
!5
0
5
x 10!3
x[4]
2
t [s]
Area 4
0 20 40 60 80!0.01
0
0.01
x[5]
2
t [s]
Area 5
Frequency deviation is controlled to zero System is first regulated to steady-state and then to the origin
41
zi =�
j�Ni
Ai jzi + Bivi + Li�PLi
0 10 20 30 40 50 60 70 80!0.2
!0.15
!0.1
!0.05
0
0.05
x[1]
3
t [s]
0 10 20 30 40 50 60 70 80
!0.3
!0.2
!0.1
0
0.1u
[1]
t [s]
�PL1 = �0.15,�PL3 = 0.05
Computational example – Area Generation Control
1"
2" 3"
4"
5"
• Four power generation areas with load frequency control • Model linearized around equilibrium (Saadat, 2002; Riverso, et al. 2012) Goals: - Restore frequency, follow load change - Allow fifth area to join the network
42
zi =�
j�Ni
Ai jzi + Bivi + Li�PLi
�PL1 = �0.15,�PL3 = 0.05
Terminal set sizes change dynamically
30 35 40 45 50 55 600
1
2
3
4
5x 10
!5
t [s]
![i]
![1]
![2]
![3]
![4]
![5]
Summary – Distributed MPC • Structured Lyapunov functions and dynamic invariant sets guarantee
stability and invariance by design
• Synthesis and control via distributed optimization [Conte, et al., ACC 2012], [Conte, et al., CDC 2012]
• Extension to Robust Tube-based MPC and Tracking MPC
[Conte, et al., ECC 2013, Conte, et al., CDC 2013, submitted]
• Plug and Play MPC enables network changes during closed-loop operation [Zeilinger, et al., CDC 2013, submitted]
43
Distributed and Real-time MPC Established approach: • Optimality • Terminal cost and constraint
Centralized MPC theory: J Recursive constraint satisfaction J Stability by design
Outline (Part II): Stability with larger region of attraction based on local information
Distributed MPC: • Reduced conservatism through
distributed optimization • BUT: Global terminal conditions
Outline (Part I): Stability and constraint satisfaction for any real-time constraint
Real-time MPC: • Flexibility and fast convergence
through interior-point methods • BUT: Variable solve-times
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